Abstract:

Given an algebraic situation described by $n\geq 1$ variables and depending on $r\geq 1$ independent parameters, the Hilbert property makes it possible to specialize the parameters and preserve the structure of the situation. The classical application, in the situation of $n=1$ variable reduces the Inverse Galois Problem to the search of geometric Galois covers of the line defined over the rationals. The main part of the talk will be devoted to the less classical situation of several variables ($n>1$). We will explain how it has led to recent progress in several topics: arithmetic Bertini theorems, polynomial versions of the Schinzel Hypothesis, the dimension growth conjecture, etc.